HF equations

The HF equations must be solved iteratively beoause the J- and K. operators in F depend on the orbitals ( ). for whioh solutions are sought. Typioal iterative sohemes begin with a guess for those ([). that appear in T", whioh then allows f to be fonned. Solutions to = e.. are then found, and those (j). whioh possess the spaoe and... 

As formulated above in terms of spin-orbitals, the Hartree-Fock (HF) equations yield orbitals that do not guarantee that P possesses proper spin symmetry. To illustrate the point, consider the form of the equations for an open-shell system such as the Lithium atom Li. If Isa, IsP, and 2sa spin-orbitals are chosen to appear in the trial function P, then the Fock operator will contain the following terms ... 

Before addressing head-on the problem of how to best treat orbital optimization for open-shell species, it is useful to examine how the HF equations are solved in practice in terms of the LCAO-MO process. 

The HF equations F ( )i = 8i (jii comprise a set of integro-differential equations their differential nature arises from the kinetic energy operator in h, and the coulomb and exchange operators provide their integral nature. The solutions of these equations must be... 

In the most eommonly employed proeedures used to solve the HF equations for non-linear moleeules, the d >i are expanded in a basis of funetions X i aeeording to the LCAO-MO proeedure ... 

All the early work was concerned with atoms, with Sir William Hartree regarded as the father of the technique. His son, Douglas R. Hartree, published the definitive book, The Calculation of Atomic Structures, in 1957, and in this he derived the atomic HF equations and described numerical algorithms for their solution. Charlotte Froese Fischer was a research student working under the guidance of D. R. Hartree, and she published her own definitive book. The Hartree—Fock Method for Atoms A Numerical Approach in 1977. The Appendix lists a number of freely available atomie structure programs. Most of these can be obtained from the Computer Physics Communications Program Library. 

Solution of the numerical HF equations to full accuracy is routine in the case of atoms. We say that such calculations are at the Hartree-Fock limit. These represent the best solution possible within the orbital model. For large molecules, solutions at the HF limit are not possible, which brings me to my next topic. 

Suppose then that we have successfully solved the HF equations for a molecule with 2m electrons as shown in Figure 6.1, and that the occupied orbitals have energies a, b, > m- The LCAO coefficients are also collected in the column vectors a, b,..., m and the electronic energy is given by... 

Ab initio ECPs are derived from atomic all-electron calculations, and they are then used in valence-only molecular calculations where the atomic cores are chemically inactive. We start with the atomic HF equation for valence orbital Xi whose angular momentum quantum number is 1 ... 

Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

The Kohn-Sham equations look like standard HF equations, except that the exchange term is replaced with an exchange-correlation potential whose form is unknown. 

In both position and momentum spaees, iterative procedures are necessary to solve the HF equations. Starting from atrial orbital (])i(0)(p), an approximate orbital, < )i( +D(p),is obtained after k+1 iterations from Eq. 13 rewritten as ... 

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

In practice, unfortunately, not even the HF equations can be solved precisely due to the complicated shapes that the orbitals assume for general low-symmetry molecular environments. Hence, one introduces a fixed basis set ip that is used to expand the HF orbitals ... 

The benefit is now that the HF equations are turned from complicated integro-differential equations into pseudo-eigenvalue equations for the unknown expansion coefficients c. 

The basis functions are most commonly chosen such that the spin-function is either a pure spin-up function a(cr) or a pure spin-down function )S(cr). They are defined such that a( ) = / ( j) = 1 and zero for any other argument cr. Since the BO and, consequently, the Fock operator do not contain any spin-dependent terms, the HF equations divide into spin-up and spin-down equations ... 

The connection to HF theory has been accomplished in a rather ingenious way by Kohn and Sham (KS) by referring to a fictitious reference system of noninteracting electrons. Such a system is evidently exactly described by a single Slater determinant but, in the KS method, is constrained to share the same electron density with the real interacting system. It is then straightforward to show that the orbitals of the fictitious system fulfil equations that very much resemble the HF equations ... 

The synthesis of group 4 alkoxide complexes grafted on the surface of silica illustrates this approach.44-47 Two routes were considered, which are schematically represented in Equations(2) and (3) they differ in the nature of the precursor complex, which is either the tetra-alkoxide M(OR)4 (Equation(2)), or the tetra-alkyl complex MR4, M = Ti, Zr, Hf (Equation(3)) ... 

Since molecular HF equations cannot be solved numerically each m.o. is expanded as an LCAO and the expansion coefficients optimized variationally. The equations in the form... 

These constitute the DFT equivalence of the HF equations and, like those, must be solved iteratively in a self-consistent (SCF) procedure. The total energy can then be obtained via the density by use of the equations above, or - again by analogy with the HF equations - by use of ... 

The occupied single-particle functions (j) and the virtual single-particle functions ( )a are solutions of the corresponding canonical HF equations... 

The HF GS density nGs(f), occurring in Eq. (64), coincides with the density calculated according to Eq. (29) from [the solutions of the HF equations (33)], because the two ways of calculation of s at equivalent. By adding and subtracting the noninteracting kinetic energy functional to HF functional in... 

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